In their celebrated work, Capelli Itzykson and Zuber established an ADE-classification of modular invariant CFTs with chiral algebra $\mathfrak{su}(2)_k$.

How much of that classification can one see using the tools of perturbative quantum field theory?

Presumably, one can't see the exceptional $E_6$, $E_7$, $E_8$ family...
what about the $A_n$ versus $D_n$ families?


2 Answers 2


The following facts might be useful to OP. Firstly, the simple simply-laced $ADE$ Lie groups are in one-to-one correspondence with finite subgroups $\Gamma$ of $SU(2)$. Secondly, for $c=1$ CFTs, the $ADE$ classification is realized by orbifold models based upon modding out string propagation on a $SU(2)$ group manifold by its finite subgroups $\Gamma$, see Ref. 1 and Ref. 2, Chap.8. Thirdly, models of the $\hat{sl}(2)$ affine Kac-Moody algebra are discussed in Ref. 2, Chap.9.

Some References:

  1. P. Ginsparg, Curiosities at $c = 1$, Nucl. Phys. B 295 (1988) 153.

  2. P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028.

  • $\begingroup$ Very useful indeed. I'm in the process of learning all that stuff, and references to good, comprehensive sources are always welcome. This does not, however, address the question: the theories that I mentioned above have central charge $c=3(h-2)/h$, where $h$ is the (dual) Coxeter number. $\endgroup$
    – André
    Commented Nov 26, 2011 at 19:34
  • $\begingroup$ I updated the answer with a pointer to Chapter 9 in Ginsparg's lectures. $\endgroup$
    – Qmechanic
    Commented Nov 26, 2011 at 20:48
  • $\begingroup$ The answer here is talking about another ADE classification in CFT, which comes from the orbifolds of $SU(2)_1$ that are given by finite subgroups of $SU(2)$ (or $SO(3)$) and which are in one-to-one correspondence with ADE graphs by the McKay correspondence. $\endgroup$
    – Marcel
    Commented Nov 27, 2011 at 12:02

The $A_n$ family is obtained by considering the usual WZW action for $G=SU(2)$ and $k=n-1$. A well-known example is the free boson at $R=\sqrt{2}$, it corresponds to an $A_2$ model: its partition function is a sum of characters for spin-0 and spin-1/2 representations.

The $D_{2\rho+2}$ and $D_{2\rho+1}$ families correspond to twisting the above theories by the non-trivial outer automorphism of $\hat{A}_1$, it turns out you can do that only for even levels.

Besides Ginsparg's lectures, you can take a look at chapter 17 in Di Francesco et al. book.


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